Math and science::Algebra::Aluffi

# Groups. Some properties of order.

### Property 1

Suppose that $$g^2 = e$$ for all elements $$g$$ in group $$G$$. Then we can say that $$G$$ is commutative.

### Property 2

Suppose $$g$$ is an element with odd order. Then $$|g^2| = |g|$$?

### Property 3

The order of $$[m]_n$$ in $$\mathbb{Z}/n\mathbb{Z}$$ is 1 if $$n | m$$, and more generally:

[$| [m]_n| = \frac{?}{\quad ? \quad }$ ]

### Property 4

Following from property 4:

• The class $$[m]_n$$ generates $$\mathbb{Z}/nZ$$ iff [what?].
• the order of every element of $$\mathbb{Z}/n\mathbb{Z}$$ [what can be said here?].